Dependence of the heavily covered point on parameters

TL;DR

This paper investigates how the heavily covered point, identified via Gromov's method, depends on parameters, using homological techniques to establish a form of continuous dependence and derive related results.

Contribution

It introduces the concept of homological continuous dependence for heavily covered points and provides elementary proofs for some corollaries.

Findings

Homological methods enable analysis of parameter dependence.

Elementary proof for a key corollary.

Insights into Gromov's method of selecting heavily covered points.

Abstract

We examine Gromov's method of selecting a point "heavily covered" by simplices formed by a given finite point sets, in order to understand the dependence of the heavily covered point on parameters. We have no continuous dependence, but manage to utilize the "homological continuous dependence" of the heavily covered point. This allows us to infer some corollaries in a usual way. We also give an elementary argument to prove the simplest of these corollaries.